Embedded newform 115.5.h.a.11.2

• Label: 115.5.h.a.11.2
• Level: $115$
• Weight: $5$
• Character: 115.11
• Analytic conductor: $11.888$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	115.h (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8875457546\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(32\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			11.2
Character	\(\chi\)	\(=\)	115.11
Dual form			115.5.h.a.21.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.40115 + 4.11377i) q^{2} +(-1.77574 + 12.3506i) q^{3} +(17.4050 - 38.1116i) q^{4} +(-3.14987 - 10.7275i) q^{5} +(-39.4406 - 86.3628i) q^{6} +(66.0549 - 57.2369i) q^{7} +(28.0444 + 195.053i) q^{8} +(-71.6639 - 21.0424i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q - 12 q^{2} - 272 q^{4} + 166 q^{6} - 246 q^{8} - 896 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/115\mathbb{Z}\right)^\times\).

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{9}{22}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.40115	+	4.11377i	−1.60029	+	1.02844i	−0.633204	+	0.773985i	\(0.718261\pi\)
							−0.967084	+	0.254458i	\(0.918103\pi\)
\(3\)	−1.77574	+	12.3506i	−0.197305	+	1.37228i	0.614759	+	0.788715i	\(0.289253\pi\)
							−0.812064	+	0.583569i	\(0.801656\pi\)
\(5\)	−3.14987	−	10.7275i	−0.125995	−	0.429098i
							\(7\)	66.0549	−	57.2369i	1.34806	−	1.16810i	0.377841	−	0.925871i	\(-0.376667\pi\)
0.970219	−	0.242230i	\(-0.0778787\pi\)
\(11\)	−97.2685	+	151.353i	−0.803872	+	1.25085i	0.160694	+	0.987004i	\(0.448627\pi\)
							−0.964565	+	0.263844i	\(0.915009\pi\)
\(13\)	−57.0614	+	65.8523i	−0.337641	+	0.389659i	−0.899025	−	0.437897i	\(-0.855724\pi\)
							0.561384	+	0.827555i	\(0.310269\pi\)
\(17\)	302.843	−	138.304i	1.04790	−	0.478560i	0.184370	−	0.982857i	\(-0.440976\pi\)
							0.863530	+	0.504297i	\(0.168248\pi\)
\(19\)	78.3773	+	35.7937i	0.217112	+	0.0991516i	0.521002	−	0.853555i	\(-0.325558\pi\)
							−0.303890	+	0.952707i	\(0.598286\pi\)
\(23\)	71.3618	+	524.165i	0.134900	+	0.990859i
							\(29\)	−4.87166	−	10.6675i	−0.00579270	−	0.0126843i	0.906714	−	0.421747i	\(-0.138583\pi\)
−0.912506	+	0.409063i	\(0.865856\pi\)
\(31\)	27.7946	+	193.315i	0.0289225	+	0.201161i	0.999159	−	0.0409978i	\(-0.0130537\pi\)
							−0.970237	+	0.242159i	\(0.922145\pi\)
\(37\)	−549.409	+	1871.11i	−0.401321	+	1.36677i	0.472844	+	0.881146i	\(0.343227\pi\)
							−0.874165	+	0.485629i	\(0.838591\pi\)
\(41\)	−1254.14	+	368.250i	−0.746070	+	0.219066i	−0.632604	−	0.774476i	\(-0.718014\pi\)
							−0.113467	+	0.993542i	\(0.536196\pi\)
\(43\)	3240.17	+	465.867i	1.75239	+	0.251956i	0.942401	−	0.334485i	\(-0.108562\pi\)
							0.809992	+	0.586441i	\(0.199472\pi\)
\(47\)	1023.21			0.463202			0.231601	−	0.972811i	\(-0.425604\pi\)
							0.231601	+	0.972811i	\(0.425604\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.5.h.a.11.2	✓	320
23.21	odd	22	inner	115.5.h.a.21.2	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.5.h.a.11.2	✓	320	1.1	even	1	trivial
115.5.h.a.21.2	yes	320	23.21	odd	22	inner